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Heuristic Search Russell and Norvig: Chapter 4 Slides adapted from: robotics.stanford.edu/~latombe/cs121/2003/home.htm Heuristic Search Blind search totally ignores where the goals are. A heuristic function that gives us an estimate of how far we are from a goal. Example: How do we use heuristics? Heuristic Function Function h(N) that estimate the cost of the cheapest path from node N to goal node. Example: 8-puzzle 5 8 4 2 1 7 3 6 N 1 2 3 h(N) = number of misplaced tiles =6 4 5 6 7 8 goal Heuristic Function Function h(N) that estimate the cost of the cheapest path from node N to goal node. Example: 8-puzzle 5 8 4 2 1 7 3 6 N 1 2 3 h(N) = sum of the distances of every tile to its goal position 4 5 6 =2+3+0+1+3+0+3+1 7 8 = 13 goal Greedy Best-First Search Path search(start, operators, is_goal) { fringe = makeList(start); while (state=fringe.popFirst()) { if (is_goal(state)) return pathTo(state); S = successors(state, operators); fringe = insert(S, fringe); } return NULL; } Order the nodes in the fringe in increasing values of h(N) Robot Navigation Robot Navigation f(N) = h(N), with h(N) = Manhattan distance to the goal 8 7 7 6 5 4 5 4 3 3 2 6 7 6 8 7 3 2 3 4 5 6 5 1 0 1 2 4 5 6 5 4 3 2 3 4 5 6 Robot Navigation f(N) = h(N), with h(N) = Manhattan distance to the goal 8 7 7 6 5 4 5 4 3 3 2 6 77 6 8 7 3 2 3 4 5 6 5 1 00 1 2 4 5 6 5 4 3 2 3 4 5 6 Properties of the Greedy Best-First Search If the state space is finite and we avoid repeated states, the best-first search is complete, but in general is not optimal If the state space is finite and we do not avoid repeated states, the search is in general not complete If the state space is infinite, the search is in general not complete Admissible heuristic Let h*(N) be the cost of the optimal path from N to a goal node Heuristic h(N) is admissible if: 0 h(N) h*(N) An admissible heuristic is always optimistic 8-Puzzle 5 8 1 2 3 6 4 2 1 4 5 7 3 N 6 7 8 goal • h1(N) = number of misplaced tiles = 6 is admissible • h2(N) = sum of distances of each tile to goal = 13 is admissible • h3(N) = (sum of distances of each tile to goal) + 3 x (sum of score functions for each tile) = 49 is not admissible A* Search A* search combines Uniform-cost and Greedy Best-first Search Evaluation function: f(N) = g(N) + h(N) where: g(N) is the cost of the best path found so far to N h(N) is an admissible heuristic f(N) is the estimated cost of cheapest solution THROUGH N 0 < c(N,N’) (no negative cost steps). Order the nodes in the fringe in increasing values of f(N) Completeness & Optimality of A* Claim 1: If there is a path from the initial to a goal node, A* (with no removal of repeated states) terminates by finding the best path, hence is: complete optimal 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 3+3 1+5 2+3 3+4 5+2 goal 0+4 3+2 1+3 2+3 5+0 3+4 1+5 2+4 4+1 Robot Navigation f(N) = g(N)+h(N), with h(N) = Manhattan distance to goal 8+3 8 7+4 7 6+3 6+5 6 5+6 5 4+7 4 3+8 3 2+9 2 3+10 3 4 7+2 7 6+1 6 5 5+6 5 4+7 4 3+8 3 3 2+9 2 1+10 1 0+11 0 1 6 5 2 4 7+0 7 6+1 6 8+1 8 7+2 7 6+3 6 5+4 5 4+5 4 3+6 3 2+7 2 3+8 3 4 5 5 6 Robot navigation f(N) = g(N) + h(N), with h(N) = straight-line distance from N to goal Cost of one horizontal/vertical step = 1 Cost of one diagonal step = 2 Consistent Heuristic The admissible heuristic h is consistent (or satisfies the monotone restriction) if for every node N and every successor N’ of N: N c(N,N’) h(N) c(N,N’) + h(N’) N’ h(N’) (triangular inequality) h(N) 8-Puzzle 5 8 1 2 3 6 4 2 1 4 5 7 3 N 6 7 8 goal • h1(N) = number of misplaced tiles • h2(N) = sum of distances of each tile to goal are both consistent Claims If h is consistent, then the function f along any path is non-decreasing: N f(N) = g(N) + h(N) f(N’) = g(N) +c(N,N’) + h(N’) h(N) c(N,N’) + h(N’) f(N) f(N’) c(N,N’) N’ h(N) h(N’) If h is consistent, then whenever A* expands a node it has already found an optimal path to the state associated with this node Avoiding Repeated States in A* If the heuristic h is consistent, then: Let CLOSED be the list of states associated with expanded nodes When a new node N is generated: If its state is in CLOSED, then discard N If it has the same state as another node in the fringe, then discard the node with the largest f Complexity of Consistent A* Let s be the size of the state space Let r be the maximal number of states that can be attained in one step from any state Assume that the time needed to test if a state is in CLOSED is O(1) The time complexity of A* is O(s r log s) Heuristic Accuracy h(N) = 0 for all nodes is admissible and consistent. Hence, breadth-first and uniformcost are particular A* !!! Let h1 and h2 be two admissible and consistent heuristics such that for all nodes N: h1(N) h2(N). Then, every node expanded by A* using h2 is also expanded by A* using h1. h2 is more informed than h1 h2 dominates h1 Which heuristic for 8-puzzle is better? Iterative Deepening A* (IDA*) Use f(N) = g(N) + h(N) with admissible and consistent h Each iteration is depth-first with cutoff on the value of f of expanded nodes f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 4 Cutoff=4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 4 Cutoff=4 4 6 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 4 Cutoff=4 4 5 6 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 5 4 Cutoff=4 4 5 6 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 6 5 4 5 6 6 4 Cutoff=4 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 4 Cutoff=5 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 4 Cutoff=5 4 6 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 4 Cutoff=5 4 5 6 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 4 Cutoff=5 4 5 7 6 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 4 Cutoff=5 5 4 5 7 6 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 4 Cutoff=5 5 4 5 7 6 6 5 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles 8-Puzzle 4 Cutoff=5 5 4 5 7 6 6 5 About Heuristics Heuristics are intended to orient the search along promising paths The time spent computing heuristics must be recovered by a better search After all, a heuristic function could consist of solving the problem; then it would perfectly guide the search Deciding which node to expand is sometimes called meta-reasoning Heuristics may not always look like numbers and may involve large amount of knowledge When to Use Search Techniques? The search space is small, and There is no other available techniques, or It is not worth the effort to develop a more efficient technique The search space is large, and There is no other available techniques, and There exist “good” heuristics Summary Heuristic function Greedy Best-first search Admissible heuristic and A* A* is complete and optimal Consistent heuristic and repeated states Heuristic accuracy IDA*